Viscous fingering instabilities in spontaneously formed blisters of MoS2 multilayers

The viscous fingering in the Hele-Shaw cell can be suppressed by replacing the upper-bounding rigid plate with an elastic membrane. Recently, graphene multilayers while polymer-curing-induced blistering showed the dynamical evolution of viscous fingering patterns on a viscoelastic substrate due to their thickness-dependent elasticity. Under certain conditions, the elastic solid-based instability couples with the viscoelastic substrate-based instability. The mechanisms underlying such a coupling in the blisters of 2D materials and the dynamical evolution of the viscous fingering patterns underneath the blisters are yet to be addressed. Herein, we investigate the viscous fingering instabilities in spontaneously formed blisters of MoS2 multilayers, and provide thorough analytical and experimental insights for the elucidation of the dynamical evolution of the viscous fingering patterns and the coupled instabilities in the blisters. We also estimate the interfacial adhesion energy of the MoS2 flakes over a (poly)vinyl alcohol (PVA) substrate and the confinement pressure inside the MoS2 blisters using a conventional blister-test model. It is observed that the presence of instability gives rise to anomalies in the modeling of the blister test. The adhesion mechanical insights would be beneficial for fundamental research as well as practical applications of 2D material blisters in flexible optoelectronics.


Introduction
6][7][8][9] Knowing the physics and chemistry underlying the intended [10][11][12] or spontaneous formation 3,[13][14][15] of blisters is extremely important for the utilization or removal of 2D material blisters. 16,17It has also been demonstrated that the blisters of direct band gap semiconductors (e.g.9][20][21] In addition, the blisters show remarkable photo-detection capability as the excitons have a longer lifetime against recombination due to the funneling effect. 22][25] The blistering of 2D materials over rigid solid substrates has been extensively investigated. 11,26However, less attention has been paid to the 2D material blister formation over so polymeric substrates, possibly due to their considerable roughness and deformability.It's interesting to note that a hydrophilic polymer PVA acts as a potential substrate for mechanically exfoliated 2D materials, offering a larger deposition yield and monolayer area. 27,28It has been observed that the 2D material blister formation over the PVA substrate may lead to an unconventional phenomenon, i.e., 'viscous ngering'. 3This type of viscous ngering matches well with the Saffman-Taylorlike instability 29 in which the length-scale of the polymeric ngers depends on the viscous and surface tension forces, elastic properties of the material, and the thickness of the conned viscoelastic lm.6][37] However, researchers have been unable to demonstrate a dynamic evolution of viscous ngering patterns using a single material.Recently, M. Pandey et al. 3 showed for the rst time that the graphene multilayer, having thicknessdependent elasticity, can show the evolution of viscous ngering patterns in a cold mist adsorption-assisted PVAcuring-induced blistering process.The adsorbed ice-water droplets over the PVA substrate locally manipulate the viscosity of the polymeric substrate surface, which alters its adhesive properties, and thereby makes the polymeric surface more deformable.The water vapor displaces the viscoelastic PVA at a raised temperature, and results in the formation of viscous ngering patterns underneath the 2D material blisters.The phase-transition of the conned matter inside a graphene blister induces hoop compression, whose suppression results in the formation of wrinkles around the perimeter of the blister or the tent at the center of the blister.The wrinkling is more prominent for blisters of thin 2D akes (single to few layers) whereas the tent formation is more pronounced for thick 2D multilayers. 32][43] The hoop compression not only causes elastic solidbased instabilities but also has an impact on the viscoelastic substrate-based instabilities.M. Pandey et al. 3 showed how the tent-like instability interacts with the viscous ngering instability underneath a multilayer graphene blister.It follows that it is quite possible for the viscous ngering instability at the interface to interact with the wrinkling instability developed around the perimeter of a blister of a relatively thin 2D elastic nanosheet.Such an investigation based on the 2D material blisters is currently lacking in the literature.However, through theoretical modeling, D. Pihler-Puzović et al. have demonstrated this interaction between the two types of instabilities in an elastic-walled circular Hele-Shaw cell. 44he onset of elastic solid or viscoelastic substrate-based instabilities and their mutual interaction depend on the strength of uid-structure interactions, which is measured by the uid-structure interaction parameter I , i.e., the ratio of viscous stresses in the viscous uid to the bending stiffness of the upper-bounding elastic membrane. 44,45In the cold mist adsorption-assisted PVA-curing-induced blistering of graphene multilayers, the water-vapor displaces the viscoelastic PVA in the in-plane direction inhomogeneously and simultaneously bends the elastic membrane in the out-of-plane direction, thereby forming the viscous ngering patterns underneath the blisters. 3At lower values of I , where the viscoelastic substrate exhibits smaller resistance to deformation than the upperbounding elastic sheet, the viscous ngering instability predominates over the elastic solid-based instability.Chopin et al. 43 demonstrated that the parameter h/s (ratio of blister's height to ake thickness) regulates the shape prole of the blisters of a plastic sheet conning a liquid.M. Pandey et al. 3 showed that the parameter h/s not only regulates the solidbased instability or the shape prole of a multilayered 2D material blister but also the viscous ngering instability underneath the blisters over a viscoelastic substrate.For a single or few-layered 2D material blister, the solid-based instability occurs in the form of periodic wrinkles around the perimeter of the blister. 38Therefore, it is essential to comprehend the involved mechanism and the factors controlling the interaction between wrinkling and viscous ngering in the blisters.In the present work, we performed the polymer-curinginduced blistering of MoS 2 multilayers under different synthesis and processing conditions to investigate the stability of the blisters and the interface based on the outcome of the blistering process.We utilize the 2D material blister-test model in the framework of 'Föppl von Kármán theory of thin elastic plates' and the two-phase uid ow model in the framework of 'simplied lubrication theory' to elucidate the role of the interfacial velocity of the viscoelastic polymer while blistering and the interfacial adhesion strength of the 2D akes with the substrate in the evolution of viscous ngering patterns underneath the blisters.We also provide intriguing insights into a scenario, where the viscous ngering instability couples with the wrinkling instability in the MoS 2 blisters.We show that the presence of a solid-or substrate-based instability results in an anomaly in the modeling, where the parameter h/s of the blisters attains unconventionally high values.

Results and discussion
We analyzed our previous experimental ndings on the PVAcuring-induced blistering of MLG akes 3,9 and observed that the conventional PVA-curing-induced blistering process results in 2D material blisters of lower h/s with a stable and regular interface whereas the cold mist adsorption-assisted blistering process results in the blisters of larger h/s with a complex interface.We further investigated the PVA-curing-induced blistering of MoS 2 akes under different synthesis and processing conditions to nd the crucial parameter that is directly connected to the synthesis and processing conditions used in the experiment, whether it be the conventional or cold mist adsorption-assisted blistering process.M. Pandey et al. 9 found that multilayered graphene blisters, formed spontaneously through the conventional PVA-curing-induced blistering process, follow the elastic plate model, which satises the condition h/s ( 1.5, owing to signicant bending rigidity of the multilayered akes having minimal interlayer slippage.We nd this observation to be also valid for multilayered MoS 2 blisters formed through the same blistering process.But interestingly, when the PVA-coated Pyrex substrate is exposed briey to a humid atmosphere at a lower temperature prior to the simultaneous curing and exfoliation step, the blisters of MoS 2 multilayers form spontaneously with an exceptionally high h/s ratio. 3The existing literature suggests that the parameter h/s is larger (>2) for the blisters following the membrane prole. 46his hypothesis seems to be oversimplied because even if the condition h/s ( 1.5 is no longer true, the blisters follow the nonlinear elastic plate prole, i.e., which is a fourth-order function of the radial distance r.The height prole of the blisters is tted to a fourth-order polynomial function of radial distance r as, w(r .We attribute the larger h/s to the interlayer sliding due to weak vdW forces in between the layers of a multilayered 2D material.Therefore, we can reasonably assume that each layer bends independently

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while blistering such that the out-of-plane bending rigidity effectively scales linearly with the number of layers, i.e., B eff ∼ N (not N 3 ). 1,47However, the level of interlayer sliding is more prominent (i) during the collapse of the blister due to the phasetransition of the conned matter, which gives rise to the tentlike shape of the blister, and also (ii) during the ice-water adsorption-assisted PVA-curing-induced blistering due to the ultralubricated interface resulting from the wetting. 3The interfacial adhesion energy density (G) of the PVA-supported MoS 2 akes consisting of N layers can be estimated using the equation: 9 where E 2D = Es is the 2D elastic stiffness of the 2D ake, s = Nt is the thickness of the 2D ake, N is the number of layers, t is the thickness of the monolayer, E is the in-plane elastic modulus of the 2D material, is the constant prefactor, n is Poisson's ratio, g w is the surface tension of water = 72 mN m −1 , q m is the water contact angle with the 2D ake, and q s is the water contact angle with the substrate.For MoS 2 blisters over the PVA substrate, we use the following parameters: 9,24,48,49 E = 270 GPa, t = 0.65 nm, n = 0.29, q m = 69°, q s = 51°.Hencky's model gives a relation between the connement pressure (Dp) and the topography of a blister, 46 as where p 0 = 1 atm.pressure = 101 325 Pa, p is the net pressure inside the blister, and K(n) is a constant prefactor; K(n = 0.16) = 3.09 for graphene, and K(n = 0.29) = 3.54 for MoS 2 . 24,48,50On employing the elastic plate model with the modied bending stiffness term for the MoS 2 blisters formed through the conventional PVA-curing-induced blistering process, the interfacial adhesion energy (G) of the MoS 2 multilayer (see Fig. 1) is estimated as ∼71.3 mJ m −2 , and the net pressure inside the nanoblisters is ∼0.15MPa.By employing the PVA-curing-induced blistering process aer exposing the PVA surface to cold mist, we observed that the ratio of blister height to ake thickness (h/s) dramatically increases, apart from observing the viscous ngering instability at the interface (see Fig. 2).It is clear from our observations that the elastic plate model that is applicable for h/s ( 1.5 is violated in spontaneously formed blisters having at least one kind of instability, either the elastic solid-based instability (tenting or wrinkling), resulting from the phase transition of the conned matter, or the viscoelastic substrate-based instability (viscous ngering). 3,38To understand the underlying mechanisms, we employ the 2D material blistering model in the framework of the 'Föppl von Kármán theory of thin elastic plates' and the 'simplied lubrication theory model' for the two-phase uid ow in an elastic-walled circular Hele-Shaw cell. 3,45The degree of instability in blistering is governed by the uid-structure interaction parameter (dimensionless), i.e., where g is the viscous stiffness in the viscous uid, and B eff is the bending stiffness of the upper-bounding elastic membrane.
The parameter I has an impact on the growth dynamics of viscous ngering patterns underneath the 2D material blisters.
The parameter I has two extreme limits: (i) I /0, which depicts the case where the viscoelastic substrate is easily deformable in comparison to the upper-bounding elastic membrane; this condition is responsible for the onset of viscous ngering instability; and (ii) I /N, which depicts the case where the upper-bounding elastic membrane is easily deformable in comparison to the viscoelastic substrate; this condition is responsible for the onset of solid-based instability.
For a constant viscosity of the viscoelastic substrate, the parameter I depends on the blister's length scale L and the parameter h/s at a unit time interval. 3The uid-structure interaction parameter I is related to the pressure for ux-driven ow P, as Paper Nanoscale Advances Assuming the ux-driven pressure to be of the order of the connement pressure inside the 2D material blister, i.e., P ∼Dp.This yields an effective uid-interaction parameter, as Assuming, L $ 2a, and using eqn (3), we obtain This approximation is valid for the case when the interlayer sliding is prominent due to weak vdW interactions, which is possible for an ultralubricated (frictionless) interface.Each layer of the multilayered ake attains the same bulging height h as that of the ake itself because each layer of the 2D material bends independently.Therefore, the effective thickness of the multilayered ake would be equal to the thickness of the single layer. 38e also observed the concurrence of wrinkling at the perimeter of blisters of thinner MoS 2 akes (N ( 100) and the viscous ngering at the interfaces (see Fig. 3).The material blisters form at a raised temperature of ∼100 °C due to the trapping of water vapor resulting from the evaporation of water content (absorbed/adsorbed) of the PVA.The blister is of a circular shape at the raised temperature due to symmetrically distributed pressure across the blister walls, and it collapses at the edge due to condensation of the water vapor (gas) into the liquid phase as it is cooled down to room temperature. 3We observe three regions in a blister, as shown in Fig. 3(e), viz.(i) region I: the spherical or tent-like tip of the blister, (ii) region II: viscous ngering patterns at the periphery of the blister, and (iii) region III: the nearly deated region of the initially circular blister due to phase transition-induced collapse.The tip height of the blister is almost 25 times larger than the average height of the wrinkles at the periphery of the blister (see Fig. 3(b)).On removing the bulged upper-bounding ake using a Nitto tape, the polymeric ngers formed at the periphery of a 2D material blister can be visualized (see Fig. 4(e)-(h)).It is clear from the topographic images (Fig. 4(g) and (h)) that the viscoelastic polymer PVA is radially displaced by the water vapor at the raised temperature.The polymeric ngers are captured by the wrinkles only at the edge of the blister, which depicts the interfacial nature of viscous ngering.The heat treatment (at ∼200 °C) of a PVA-supported 2D material blister in upside-down orientation can also reveal the type of viscous instability whether interfacial or bulk. 3Because of good thermal conductivity of the 2D material, the polymeric ngers at the periphery of the blister deform prior to the central region of the blister, depicting the interfacial nature of the viscous ngering.
Interestingly, we found that the wrinkle wavelength is nearly equal to the nger wavelength (see Fig. 3(d) and (f)).Therefore, the parameters governing the wrinkling in the thin MoS 2 blister are directly related to the parameters for the onset of viscous ngering.Finding a critical criterion for such a situation to occur would indeed be worthwhile.According to the 'local wavelength law', the wrinkle wavelength is given by 38 where K eff is the 'effective stiffness', i.e., the spring constant per unit area of a compliant substrate, having units of ½energy ½length 4 .The wavelength of the polymeric ngers, derived by Saffman and Taylor, is given by 44,45 where U is the instantaneous radial velocity of the circular interface, s is the surface tension at the water vapor to On setting l w z l f , we obtain where K eff z g L 4 manifests the energetic cost required to deform the viscoelastic substrate. 38However, in the case of blistering of a 2D elastic membrane over a 2D layered vdW For the volumetric ow rate , and for a given system with xed elastic and viscoelastic parameters, we obtain for all time: where C is a constant prefactor.Eqn (12) implies that a resonance between the in-plane velocity of the radially propagating viscoelastic uid and the out-of-plane debonding velocity of the upper-bounding membrane leads to a condition where both the viscous ngering and the wrinkling instability in a blister interact with each other and occur concurrently.For the uxdriven pressure to be of the order of the connement pressure, i.e., P ¼ 6mQ pb 3 $ Dp; I /I *0U/U*.It is to be noted that the atmospheric pressure outside the blister (i.e., p 0 ) remains constant, which has not been taken into account in the model to avoid complexities.This hypothesis effectively helped us in relating the growth dynamics of the viscous ngering patterns with the interfacial adhesion energy (G) and the effective interfacial in-plane velocity (U*) of the viscoelastic PVA being displaced by the water vapor while blistering (see Fig. 5).
We observed that both the parameters (G and U*) simultaneously affect the growth dynamics of the ngering patterns.The 2D material blisters with stable interfaces form spontaneously in the conventional PVA-curing-induced blistering process, where the stability of the interface is due to the higher viscosity and stronger viscous stresses of the PVA surface.The viscoelastic PVA remains typically static while PVA-curinginduced blistering under ambient conditions (see Fig. 1), however, under the cold-mist adsorption-assisted PVA-curinginduced blistering, the viscoelastic PVA radially propagates with an appreciable in-plane velocity and makes the interface unstable (see Fig. 5).The weaker the interfacial adhesion of the multilayered MoS 2 akes, the larger the polymeric nger length due to the smaller interfacial velocity of the viscoelastic PVA.The ice-water droplets locally reduce the viscosity of the PVA and make it quite deformable at raised temperatures (∼100 °C).The adsorbed water content or wetting results in the formation of blisters with exceptionally large values of the parameter h/s, which indicates signicant interlayer sliding during the blistering of the 2D multilayered ake.It should be noted that the blisters with a large h/s [ 1.5 still follow the elastic plate prole.This observation suggests that the parameter h/s is signicantly inuenced by the interlayer slippage as well as the interfacial adhesion strength.

Conclusion
In summary, the synthesis and processing conditions, in addition to the interfacial debonding strength of the MoS 2 multilayers, the connement pressure inside the blisters, and the phase of the conned matter have crucial roles to play in the stability and dynamics of the blister system.The elastic properties of the 2D material, the viscoelastic properties of the

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polymeric substrate, and the physical properties of the conned uid affect the blistering of the 2D multilayers.The adsorption of ice-water droplets on the hydrophilic PVA surface favors the interlayer sliding while blistering of the 2D multilayers at a raised temperature, which results in the exceptionally high values of the parameter h/s, violating the non-linear elastic plate model.In the cold mist adsorption-assisted blistering of a 2D multilayer, the lower the interfacial radial velocity of the viscoelastic PVA, the larger the nger length.When the criterion of the resonance between the interfacial (in-plane) velocity of the polymer and the out-of-plane bending velocity of the 2D material is met, the solid-based wrinkling instability occurs concurrently with the viscous ngering instability.The mechanical insights of the blisters could potentially provide information about regulating the viscoelastic substrate-based instabilities in the blistering of the 2D layered vdW materials.
The developed understanding might facilitate the nextgeneration applications of 2D materials and their blisters in exible electronics, biomedical implants, single-photon detection, micro-/nanoelectromechanical sensing, etc.

PVA-curing-induced blistering of MoS 2 multilayers under different synthesis and processing conditions
The blisters of micromechanically exfoliated MoS 2 multilayers are formed by a PVA-curing-induced blistering process both with and without the adsorption of tiny ice-water droplets (mist) over the PVA-coated Pyrex substrate prior to the mechanical exfoliation step.The steps in detail can be seen elsewhere. 3

Characterization
The identication of MoS 2 blisters is carried out using interference reection microscopy (IRM).The Raman and PL spectra have been acquired from the identied blisters using a HORIBA LabRAM HR Evolution system under ambient atmospheric conditions.The laser light output power is kept low (to prevent local heating) at ∼1 mW for the laser excitation wavelength of 532 nm using a 100× air objective lens (NA = 0.8) and a detector grating of 600 lines per mm in a confocal microscopy setup.The tapping-mode AFM measurements have been performed with a standard silicon cantilever using a Bruker MultiMode-8 AFM setup.The AFM data visualization and analysis have been carried out using the Gwyddion and WSxM soware packages.

Fig. 1
Fig. 1 (a) Nanoblisters of multilayer MoS 2 , having the number of layers N = 39, formed spontaneously through the conventional PVA-curinginduced blistering process.(b) The interfacial adhesion energy and the net confinement pressure as a function of the blister aspect ratio for 14 different nano-blisters across the flake.

Fig. 2
Fig. 2 Blisters of MoS 2 multilayers having viscous fingering instability underneath.The scale bar is 20 mm.

Fig. 3
Fig. 3 (a) AFM topographic 2D map of the blister with coupled instability; (b) AFM height profiles along the lines marked in (a).(c and d) AFM topographic 2D height image and the line profile across the periphery of the blister shown by a red dotted circle, respectively, and (e and f) optical image captured at the interface of the blister using interference reflection microscopy, depicting three zones of the blister, and the gray value plot profile across the periphery of the blister shown by a blue dotted circle, respectively.

Fig. 4
Fig. 4 Multilayer MoS 2 blister with coupled instability: (a) AFM topographic 3D image, (b) optical image (scale bar: 10 mm), (c and d) Raman and PL spectra acquired at a point (red-colored, as shown in (b)) located at the center of the blister, respectively, (e and f) AFM topographic 2D and 3D images of the viscous fingering pattern on the PVA surface, (g and h) top-view and side-view of the polymeric fingers, respectively, and (i) Raman spectrum acquired at the center of the pattern, indicating no degradation of the polymer.

Fig. 5
Fig. 5 Graphical representation of circular blisters showing the dynamical evolution of interfacial viscous fingering instability with respect to the effective interfacial radial velocity (U*) and the interfacial adhesion energy (G).The scale bar is 10 mm.